You may be inclined to avoid the overly-touristy areas, but you MUST go to the viewing decks of the Eiffel Tower. The tower itself--as well the views it provides--are incredible. Be sure to visit at sundown, the best time to experience the tower, as it becomes illuminated by thousands of lights.

To save some money in the city, shop for meals at open-air food markets. You can find breads, cheeses and pastries, and if you're renting an apartment--a great way to see the city--you can prepare your own authentically Parisian meals.

I am sure you have heard this by now but try to speak French. It is very important to the French people that you attempt to speak their language and you are not seem as an arrogant American who expects everybody to speak English.